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Mirrors > Home > ILE Home > Th. List > pwex | GIF version |
Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
zfpowcl.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
pwex | ⊢ 𝒫 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfpowcl.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | pweq 3403 | . . 3 ⊢ (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴) | |
3 | 2 | eleq1d 2151 | . 2 ⊢ (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V)) |
4 | df-pw 3402 | . . 3 ⊢ 𝒫 𝑧 = {𝑦 ∣ 𝑦 ⊆ 𝑧} | |
5 | axpow2 3970 | . . . . . 6 ⊢ ∃𝑥∀𝑦(𝑦 ⊆ 𝑧 → 𝑦 ∈ 𝑥) | |
6 | 5 | bm1.3ii 3919 | . . . . 5 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧) |
7 | abeq2 2191 | . . . . . 6 ⊢ (𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧)) | |
8 | 7 | exbii 1537 | . . . . 5 ⊢ (∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 ↔ 𝑦 ⊆ 𝑧)) |
9 | 6, 8 | mpbir 144 | . . . 4 ⊢ ∃𝑥 𝑥 = {𝑦 ∣ 𝑦 ⊆ 𝑧} |
10 | 9 | issetri 2617 | . . 3 ⊢ {𝑦 ∣ 𝑦 ⊆ 𝑧} ∈ V |
11 | 4, 10 | eqeltri 2155 | . 2 ⊢ 𝒫 𝑧 ∈ V |
12 | 1, 3, 11 | vtocl 2662 | 1 ⊢ 𝒫 𝐴 ∈ V |
Colors of variables: wff set class |
Syntax hints: ↔ wb 103 ∀wal 1283 = wceq 1285 ∃wex 1422 ∈ wcel 1434 {cab 2069 Vcvv 2610 ⊆ wss 2982 𝒫 cpw 3400 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-11 1438 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 |
This theorem depends on definitions: df-bi 115 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-v 2612 df-in 2988 df-ss 2995 df-pw 3402 |
This theorem is referenced by: pwexg 3974 p0ex 3979 pp0ex 3980 ord3ex 3981 abexssex 5803 npex 6777 axcnex 7141 pnfxr 7285 mnfxr 7289 ixxex 9050 |
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